I first heard of this art piece by reading Denis Cosgrove's essay on cartography and modern art in the 20th century. The essay is a brilliant piece of writing that expands the study of artistry and mapmaking beyond the early modern period. While the piece can be found at JSTOR, I have linked it here for your viewing pleasure.

No matter where you hear it – be it a ranting posting on a forum, a podcast by a know-it-all, an editorial in a newspaper, or the rhetoric of self-righteous politicians – the expression "America is a republic, not a democracy" is used to show that the United States was not founded on the principal of majoritarian rule, but on the principals enumerated in the Constitution.
The writings of some of the Founding Fathers seem to point towards this view.

Thomas Jefferson – the oft used source in discussion on the structure of American governance – wrote:

A democracy is nothing more than mob rule, where fifty-one percent of the people may take away the rights of the other forty-nine.

The framers of the Constitution (excluding Jefferson) sought to limit the idea of mob rule from forming in the nascent United States. James Madison, instrumental in crafting the Constitution, wrote in Federalist #10:

Hence it is that such democracies have ever been spectacles of turbulence and contention; have ever been found incompatible with personal security or the rights of property; and have in general been as short in their lives as they have been violent in their deaths. Theoretic politicians, who have patronized this species of government, have erroneously supposed that by reducing mankind to a perfect equality in their political rights, they would, at the same time, be perfectly equalized and assimilated in their possessions, their opinions, and their passions

Similar ideas written John Adams, Alexander Hamilton, and other figures crucial for the formation of the United States show that the idea of a direct democracy, in the minds of the founders, would devolve into mob rule, where the majority will undoubtedly strip away the innate rights of the minority. In this context, we need to keep in mind that the government of the United States was idealized to be a bulwark against the return of tyranny. It did not matter if the tyranny came from the King of England or from the majority rule of Americans. Democracy was a slippery slope towards authoritarianism.

The founders used the term “republic” to describe the governmental structure of the United States. Instead of direct democratic rule, this republic would instead be governed by elected representatives and appointed leaders that would guide and manage the United State in an enlightened manner. With the exception of local congressmen, major offices in the government were either appointed or elected by bodies independent of the federal government (e.g. the electoral college and state legislatures). Of those who could vote, only property-owning men of English or Scottish origin could. Needless to say, the percentage of eligible voters was small compared to the entire population of the United States.

The constitutional structure of the federal government - with its enumerated definition of powers - is contrasted with simple majority rule. At the end of the day the founding fathers sought to emulate republican Rome, not democratic Athens.

I guess this settles the debate: America is a republic, not a democracy!

...except that it's not over. Two centuries have passed since the revolution and the drafting of the constitution; many things have changed. Suffrage has been extended to all American citizens; Senators are now elected by the citizens of their states, and Presidents, while still elected by the electoral college, campaign across the country for the millions of votes needed to win the Presidential elections. With a wider electorate, the US has become much more democratic.

A thing to consider is the definition of what is a "republic" versus a "democracy" has changed in the intervening two centuries. Political scientists argue over the exact definition of the word "democracy", but they agree that democracies share similar traits with one another: free, multi-candidate elections; the ability of voters to punish elected representatives by voting for someone else into that office; and the institutional constraints on executive control. We can argue over the meaning of what is a "true" democracy, especially argue over a definition that fits our opinions of that definition. But if we to place the United States as either a democracy, an authoritarian state, or an anacracy (i.e. without an government of some import to the state), the United States is considered a democracy. This poli-sci definition doesn't make everyone happy, but it does seek to be as free from ideological biases and as objective as possible.

Then, what is a republic? The term originates after the ousting of the Tarquinii dynasty from ancient Rome and the establishment of a government free from monarchal tyranny. This res publica - "thing of the people" - was named as the power of the state did not lie with the kings of Rome but on the Senate and the two consuls elected every year by the citizens of Rome. It's perfectly understandable the American revolutionaries, fighting against monarchal tyranny, would hark back to Rome for inspiration in setting up the government. The term "republic" was naturally appropriated by the early Americans to signal a disdain towards monarchy; a repudiation of the "divine right" of rule; and an appreciation for a government based on the concept of natural law and the rights of man. The fact that a government based on high-minded principals was not democratic - democratic as we define it - was noted with irony by the early American leaders. We need to remember that the federal government was meant to be anti-tyrannical and not pro-democratic.

The hoopla this debate incites is also political in nature. A typical debate follows as such:

The more leftist/liberal side of this debate will argue for a particular policy position and cites polling and mass opinion to show popular support for this position.

The more right-wing/libertarian side would then argue that the United States was founded as a republic and not a democracy; essentially, damn public opinion.

The right-winger would then look at the befuddled face of the leftist and elaborate that the republican nature of the United States is designed to prevent rash, popular ideas from being implemented to prevent the "mob" from imposing their will on the minority(i.e. the right-wingers) and stripping away the freedom and liberty of the minority.

The leftist would then argue that the terms "republic" and "democratic" are not mutually exclusive...

...and the right-winger would then use his own definition of the word to argue against the lefty.

I admit, this is a vast oversimplification of the debate, but tends to be true. What is revealing in this debate is how self-serving the leftist and the right-winger are: the leftist and the right-winger will conveniently abandon their definitions if popular opinion or constitutional constraints are in contrast to their ideological view point. But - more important on the topic on-hand - is that both sides are arguing over two different sets of definitions and misconceptions of the founding of this country.

The United States is, by every objective definition of the word, a democracy. But this definition did not exist in 18th century America. The term "republic" is now meant to be used as short-hand for "representative democracy" - but this term was also not a construct of the 18th century. The United States was not founded as a democracy, but the United States has changed, as have the words we use to assign what America is. As long both sides of this debate are completely unaware of the differing definitions each has, the debate will be a futile, aggravating experience for all involved.

*On a side note, it doesn't help matters when Republicans (i.e. the GOP) are more likely to use this debate not only to invalidate the positions of Democrats, but the basis of the Democratic party. Republican = defenders of the Republic VS Democrats = advocates of democracy

The problem with any "debate" about the Texas textbook controversy is the illusion that there needs to be "balance" in the way history is presented; that the goal of the education system is to present "both sides" of any given issue.
In all cases of history, there is not a liberal historiography or a conservative historiography, but simply history itself. The idea that we need to "balance" history is absurd. The goal, rather, is to strive for a general, objective overview of world, national, and state history that will be taught to kids in our education system here in Texas. This not only applies to conservatives who wish to gloss over inconvenient facts that conflict with their political ideology (e.g. the secular nature of the Constitution), but also to post-modernist/deconstructionists who wish to impose a world-view hostile to empiricism and replace it with the notion of relative truths.

We need to get outside this frame of mind that celebrates the balancing of two mutually exclusive "truths" and instead focus on the idea that history, even inconvenient histories, should be taught to students despite the inevitable cries of protests of special interests groups and stiffing parents. Likewise, educators do not have the right to make moral or political claims about historical events because ideological groups will always try to indoctrinate children to support their own viewpoint on a given matter.

As a scientifically minded individual, I am always in favor of teaching kids critical thinking skills. Critical thinking, however, is always endangered when ideologues try to influence not only what is taught, but how one should think and feel about a historical matter. If given the mental toolkit in order to think critically, we need to trust kids to learn history and interpret the causes, the effects, and the morality of those events on their own and amongst themselves. If we cannot trust them to think for themselves and instead feel the need to teach them the "proper" histories, then this paternalism negates the purpose of teaching critical thinking.

In the end, however, the children and teens of Texas (and other states) will get a stilted textbook that is devoid of any interesting history that will lead most students to disdain any future references of historical matters. This bothers me more than anything else about this whole controversy.

One of the first podcast I ever listened to was "12 Byzantine Emperors" by Lars Brownsworth. The podcast tells the history of the reign of 12 different emperors (and empress) of the Eastern Roman empire, starting with Diocletian and ending with Constantine XVI. This podcast series is, without a doubt, one of the best history series out there today.
Fortuitously, I stumbled upon this lecture series by Brownsworth himself on Youtube. Very interesting.